Judicious

I have an image of a PC with OS and all apps installed. It's activated using an OEM key.

I want to install this image onto another PC. I've purchased the appropriate licenses and a Retail Boxed Windows 7 Ultimate.

Can I just install the OEM image and change the product key? Or do I have to install Windows from scratch?

Install Windows from scratch.
Besides the license issue, the installation has been done to another system, and its an OEM system at that, and all the drivers and hardware configuration are set for the PC it was installed on.
Now, you can do it, some people will mess with changing a drive with an OS to another machine. But the multitude of problems you will have to work through in order to get the installation to work on another machine is simply not worth it most of the time, and to that end, the OS, even if it seems okay at first, can turn buggy acting and all sorts of strange problems will keep popping up from time to time.

However, from a tech standpoint the above people have correct advice. Transferring a drive from one computer to a different computer most likely may not work or have tons of errors to resolve. It will probably run better if you do a clean install.